document.write( "Question 1024301: Hi my beloved tutors, I need help on solving this problem,
\n" ); document.write( "could you help me? Thankss\r
\n" ); document.write( "\n" ); document.write( "Consider the set of all the 7-digit numbers that can be
\n" ); document.write( "obtained using, for each number, all the digits 1, 2, 3,...,7.
\n" ); document.write( "List the numbers of the set in increasing order and split the
\n" ); document.write( "list exactly at the middle into two parts of the same size.
\n" ); document.write( "What is the last number of the first half?\r
\n" ); document.write( "\n" ); document.write( "(A) 1234567 (B) 3765421 (C) 4123567 (D) 4352617 (E) 4376521 \n" ); document.write( "
Algebra.Com's Answer #639766 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "There are 7! = 5040 numbers in the list.\r\n" );
document.write( "We want to know what the 5040/2 = 2520th\r\n" );
document.write( "number is. So we seek arrangement #2520\r\n" );
document.write( "in the list.\r\n" );
document.write( "\r\n" );
document.write( "7! = 7*6!. There are 7 groups of 6! beginning\r\n" );
document.write( "with each of the 7 digits.  6! = 720. Therefore,\r\n" );
document.write( "\r\n" );
document.write( "the 1st 6! begin with 1. That's #1 through #720.\r\n" );
document.write( "the 2nd 6! begin with 2. That's #721 through #1440.\r\n" );
document.write( "the 3rd 6! begin with 3. That's #1440 through #2160.\r\n" );
document.write( "the 4th 6! begin with 4. That's #2161 through #2880. \r\n" );
document.write( "\r\n" );
document.write( "So #2520 will be in the middle of the 4th 6!\r\n" );
document.write( "\r\n" );
document.write( "We examine the 6! that begin with 4.\r\n" );
document.write( "\r\n" );
document.write( "6! = 6*5!. There are 6 groups of 5! beginning\r\n" );
document.write( "with 4 which have as 2nd digit each of the 6 digits\r\n" );
document.write( "1,2,3,5,6,7.  5! = 120.  Therefore\r\n" );
document.write( "\r\n" );
document.write( "the 1st 5! begin with 41. That's #2161 through #2280.\r\n" );
document.write( "the 2nd 5! begin with 42. That's #2281 through #2400.\r\n" );
document.write( "the 3rd 5! begin with 43. That's #2401 through #2520.\r\n" );
document.write( "\r\n" );
document.write( "Aha! #2520 happens to be the very one we are looking \r\n" );
document.write( "for.  And it is the last and largest one on the list\r\n" );
document.write( "which begins with 43.  That would have to be 4376521\r\n" );
document.write( "\r\n" );
document.write( "Answer:  4376521, choice E.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );